Rayleigh scattering is the scattering of electromagnetic radiation by particles smaller than the wavelength of the radiation. The phrase is used for such scattering, and also for Rayleigh's mathematical model of scattering, which nicely fits scattering in this regime.
The blue of the sky is due to the dependence of wavelength on the amount of scattering (λ-4): shorter-wavelength light (blue) is scattered more, leaving the direct Sun looking less blue (yellowish) and the direct Sun at sunrise and sunset when the light passes through the most atmosphere as even less blue (reddish). The scattered light is polarized, such that polarized filters can attenuate the scattered blue light of the sky. The resulting scattered light shows the spectrum of the incoming light, most attenuated at the shorter (bluer) end.
Regarding astronomy, Rayleigh scattering contributes to reddening and extinction. It is also taken into account in studying atmospheres of extra-solar planets and other planets and moons through transmission spectroscopy. It is also expected to affect CMB photons, and possibly discernible in the CMB spectrum.